Question

Write the equation relating the half-life of a secondorder reaction to the rate constant. How does it differ from the equation for a first-order reaction?

Short answer

2nd-order: \( t_{1/2} = \frac{1}{k[A]_0} \); 1st-order: \( t_{1/2} = \frac{0.693}{k} \). Second-order depends on initial concentration, first-order does not.

Step-by-step solution

Understand the Concept of Half-life

Half-life (\( t_{1/2} \)) is the time it takes for the concentration of a reactant to decrease to half of its initial concentration. The equation for half-life differs between reaction orders.

Second-Order Reaction Formula

For a second-order reaction with the rate law \( Rate = k[A]^2 \), the half-life equation is derived from integrating the rate law:\[ t_{1/2} = \frac{1}{k[A]_0} \]where \( k \) is the rate constant and \( [A]_0 \) is the initial concentration of the reactant.

First-Order Reaction Formula

For a first-order reaction, the rate law is \( Rate = k[A] \). The half-life formula is:\[ t_{1/2} = \frac{0.693}{k} \]This formula shows that the half-life is independent of the initial concentration.

Compare the Equations

The key difference is that the half-life of a second-order reaction, \( t_{1/2} = \frac{1}{k[A]_0} \), depends on the initial concentration, while for a first-order reaction, \( t_{1/2} = \frac{0.693}{k} \), it does not.

Key concepts

Second-Order Reaction

A second-order reaction is a chemical reaction where the rate is proportional to the square of the concentration of one reactant. In simpler terms, the speed at which the reaction proceeds is highly dependent on how much reactant you start with. This type of reaction often follows the rate law given by \( Rate = k[A]^2 \), where \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant.
In the context of half-life, which is the time required for the reactant concentration to reach half of its original amount, a distinctive aspect is its dependence on the initial concentration. For second-order reactions, the half-life formula is \( t_{1/2} = \frac{1}{k[A]_0} \). This means that as the initial concentration \( [A]_0 \) decreases, the half-life increases.
This concentration-dependent half-life makes these reactions act slower at lower concentrations. Each time the concentration halves, the time it takes roughly doubles. Thus, understanding this principle is crucial for predicting how reactions progress, especially in industrial processes or environmental systems.

Rate Constant

The rate constant, symbolized as \( k \), plays a pivotal role in defining the speed of a chemical reaction. It essentially serves as a proportionality factor in the rate law equation and helps determine how quickly reactants are converted into products.
For different orders of reactions, the rate constant holds different implications. In a second-order reaction where the formula is \( Rate = k[A]^2 \), the unit of \( k \) is typically \( M^{-1}s^{-1} \) (where \( M \) stands for molarity, and \( s \) for seconds). Meanwhile, in first-order reactions, where the rate law is \( Rate = k[A] \), the unit of \( k \) becomes \( s^{-1} \).
Understanding the rate constant is crucial to comprehending how conditions such as temperature, catalysts, and concentration affect the reaction rate. A higher rate constant implies a faster reaction. It's important to note that while the rate constant is a fixed value for a given reaction at a specific temperature, it can be altered by changing other factors like temperature or adding catalysts.

First-Order Reaction

First-order reactions are characterized by a rate that is directly proportional to the concentration of a single reactant. Simply put, the pace of the reaction at any point is determined by how much of the reactant is left. The rate law for a first-order reaction is represented by \( Rate = k[A] \), where \( k \) is the rate constant and \( [A] \) is the concentration of the reactant.
A unique feature of first-order reactions is that the half-life remains constant regardless of the initial concentration. This characteristic is evident in the half-life formula \( t_{1/2} = \frac{0.693}{k} \). The half-life being independent of \( [A]_0 \) means it remains the same each time the concentration halves.
Such consistent half-life makes first-order reactions particularly straightforward to handle and predict—they are common in processes such as radioactive decay and some biochemical reactions. They provide a reliable model for many natural and industrial processes because their simplicity makes mathematical modeling relatively easy.